120 Proof

By the Theorem 48, the general solution to the differential equation is

$$X(t)=\exp (tA)X_0+{\int }_0^t\exp ((t-s)A)BU(s)𝑑s$$

where by hypotheses there exists $K>0$ such that

$$\parallel B\parallel \parallel U(s)\parallel \le K\mathit{\hspace{1em}\hspace{1em}}(s>0)$$

and by the Lemma 118

$$\parallel \exp (tA)\parallel \le M{e}^{-t\delta }\mathit{\hspace{1em}\hspace{1em}}(t>0),$$

so

$$\parallel X(t)\parallel \le M{e}^{-t\delta }\parallel X_0\parallel +{\int }_0^t\parallel \exp ((t-s)A)\parallel \parallel B\parallel \parallel U(s)\parallel 𝑑s$$